MA004: Intermediate Algebra

Unit 5: FunctionsThis unit will introduce you to functions in general, operations on
functions, and inverse functions. This is a very important unit
especially for students that will be continuing to calculus. You will
also learn about two very important functions: the exponential function
and its inverse, the logarithmic function. You will also study various
applications of these functions. The exponential function has many
applications. One of the most common applications is to calculate
compound interest.

Unit 5 Time Advisory
Completing this unit should take approximately 27.5 hours.

☐Subunit 5.1: 4 hours

☐
Subunit 5.2: 4.5 hours

☐
Subunit 5.3: 4.5 hours

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Subunit 5.4: 3 hours

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Subunit 5.5: 3 hours

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Subunit 5.6: 4 hours

☐
Subunit 5.7: 4.5 hours

Unit5 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- determine the domain of a function;
- evaluate functions at numerical and variable inputs;
- evaluate a sum, product, difference, and quotient of functions at
numerical and variable inputs;
- compute the composition of two functions;
- determine the inverse of a function;
- graph quadratic functions by identifying key points;
- find compound and continuous interest; and
- solve exponential and logarithmic equations.

Instructions: Read Section 10.1 in Chapter 10 of your textbook,
pages 386–392, to learn about functions and functional notation.
Functions are relations between two variables (in general denoted by
x and y, where x is the independent variable and y is the dependent
variable) such that for each value of x there is only one value of
y. Note that this reading covers the topics in Subunits
5.1.1–5.1.4.

Reading this section and taking notes should take approximately 2
hours.

Instructions: Complete pages 96–99 of Wallace’s workbook. Try to
complete this exercise before watching the videos in Subunits
5.1.1–5.1.4, and then review the worksheet as you follow along with
the videos for solutions.

Instructions: Watch the video linked above, which discusses the
definition of a function as well as the vertical line test.
Functions are relations between two variables (in general denoted by
x and y, where x is the independent variable and y is the dependent
variable) such that for each value of x there is only one value of
y. If a relationship between x and y is graphed, one can determine
if the relationship is a function by using the vertical line test.
The graph represents a function if every vertical line intersects
the graph at most once.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.1.2 Domain of a FunctionNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.1.

Instructions: Watch the video linked above, which explains the
notion of the domain of a function. The domain of this function
contains all values that can replace x such that you can calculate
y. Speaking geometrically, this means all values of x such that a
vertical line through x will intersect the graph.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.1.3 Function NotationNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.1.

Instructions: Watch the video linked above, which discusses
function notation. The explicit function notation is f(x), so you
write y = f(x), which means y is a function of x. This is a
convenient notation to explicitly define a function and evaluate.
For instance, to write y = f(3) means “what is y when x = 3?”

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.1.4 Function Evaluation at ExpressionNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.1.

Instructions: Watch the video linked above, which discusses
evaluating a function at an algebraic expression. In the last video,
you saw how to evaluate at a number, say f(3) for instance. Using
the same concept, you can evaluate at an algebraic expression, for
instance f(x+3).

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Read Section 10.2 in Chapter 10 of your textbook,
pages 393–400, to learn the algebra of functions. As in arithmetic,
you can add, subtract, multiply, and divide functions. There is an
additional but important operation called composites. A composite of
a function with another function is a function at the second
function. You write (f°g)(x) = f(g(x)). Note that this
reading covers the topics in Subunits 5.2.1–5.2.3.

Reading this section and taking notes should take approximately 2
hours.

Instructions: Complete pages 100–101 of Wallace’s workbook. Try to
complete this exercise before watching the videos in Subunits
5.2.1–5.2.3, and then review the worksheet as you follow along with
the videos for solutions.

Reading this section and taking notes should take approximately 1
hour.

Instructions: Watch the video linked above, which discusses the
arithmetic of functions. In a very natural way, you can add,
subtract, multiply, and divide functions. This video gives examples
of each.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.2.2 Add/Subtract/Multiply/Divide Functions (Part 2)Note: This subunit is also covered by the reading and assessment
assigned in Subunit 5.2.

Instructions: Read Section 10.3 in Chapter 10 of your textbook,
pages 401–405, to learn about inverse functions. Knowing the inverse
function of a function can be very helpful in solving many equations
in the same way it is helpful to understand that subtraction is the
inverse of addition and squaring is the inverse of square root. Note
that this reading also covers the topics in Subunits 5.3.1–5.3.3.

Reading this section and taking notes should take approximately 2
hours.

Instructions: Complete pages 102–104 of Wallace’s workbook. Try to
complete this exercise before watching the videos in Subunits
5.2.1–5.3.3, and then review the worksheet as you follow along with
the videos for solutions.

Instructions: Watch the video linked above, which discusses how to
determine if a function is the inverse of another function. The
inverse function undoes what a function does to a value of x. For
instance, if f(x) = x + 3, then g(x) = x - 3 is the inverse. The
test to determine the inverse is (f°g)(x) =
(g°f)(x) = x.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.3.2 Find the Inverse of a FunctionNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.3.

Instructions: Watch the video linked above, which discusses how to
find the inverse of a function. If y = f(x) is the function, then
the inverse function, denoted by f-1(x), can be found by
solving for x and then replacing y with x and x with y.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.3.3 Graph the Inverse of a FunctionNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.3.

Instructions: Watch the video linked above, which illustrates how
to graph the inverse of a function. The inverse of a function is
interchanging the role of the x and y. Graphically, this means that
the graph of an inverse function is a reflection across the line y =
x.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Read Section 9.11 in Chapter 9 of your textbook,
pages 380–384, to learn the characteristics of the graphs of
quadratics. When graphed, the quadratic function has several
characteristics. Each quadratic function graph has a vertex (a point
where the graph stops going down and starts going up or vice versa),
a line of symmetry, and a maximum value or a minimum value. Note
that this reading covers the topics in Subunits 5.4.1 and 5.4.2.

Reading this section and taking notes should take approximately 2
hours.

Instructions: Complete page 105 of Wallace’s workbook. Try to
complete this exercise before watching the videos in Subunits 5.4.1
and 5.4.2, and then review the worksheet as you follow along with
the videos for solutions.

Instructions: Watch the video linked above, which discusses the key
points of the graph of the quadratic function. This video describes
what direction the graph is (either U shaped or up-side-down U
shaped), the y-intercept, the x-intercept, and the vertex of the
graph.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.4.2 Graph Key Points of Quadratic Functions (Part 2)Note: This subunit is also covered by the reading and assessment
assigned in Subunit 5.4.

Instructions: Read Section 10.4 in Chapter 10 of your textbook,
pages 406–409, to learn about exponential functions. The exponential
function is very important for many applications, but the most
common application is for certificate of deposits calculations. Note
that this reading also covers the topics in Subunits 5.5.1–5.5.2.

Reading this section and taking notes should take approximately 2
hours.

Instructions: Complete pages 106–107 of Wallace’s workbook. Try to
complete this exercise before watching the videos in Subunits 5.5.1
and 5.5.2, and then review the worksheet as you follow along with
the videos for solutions.

Instructions: Watch the video linked above, which discusses
exponential equations with a common base. When an equation
consisting of two exponential expressions has the same base and the
unknown value is in the exponent, then the exponents must be
equal.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.5.2 Exponential Equations with Binomial ExponentsNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.5.

Instructions: Read Section 10.6 in Chapter 10 of your textbook,
pages 414–419, to learn an application of the exponential functions,
namely compound interest. A bank has certificates of deposits (CDs)
for sale. The interest rate paid is determined by the length of time
that the consumer chooses. The bank will compound the interest
(calculate the interest, and then pay interest on this interest)
during various intervals of time (quarterly, monthly, annually).
Once all of these parameters are known, the compound formula
calculates the return on your CD. Note that this reading also covers
the topics in Subunits 5.6.1–5.6.4.

Reading this section and taking notes should take approximately 2
hours.

Instructions: Complete pages 108–111 of Wallace’s workbook. Try to
complete this exercise before watching the videos in Subunits
5.6.1–5.6.4, and then review the worksheet as you follow along with
the videos for solutions.

Instructions: Watch the video linked above, which discusses an
application of the exponential function, namely compound interest.
The video gives the formula for return of money (future value of
your money) and gives an example.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.6.2 Finding the Principle (P) Given the Return Amount (A)Note: This subunit is also covered by the reading and assessment
assigned in Subunit 5.6.

Instructions: Watch the video linked above, which discusses an
application of the exponential function, namely continuous
compounding interest. Continuous compounding means the bank is
compounding every millisecond of every day. When this happens, a new
constant is introduced: e. e is a constant and is approximately
equal to 2.7182818. Then the formula becomes A = Pert.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.6.4 Finding the Principle (P) for Continuous CompoundingNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.6.

Instructions: Watch the video linked above, which asks the reverse
of the examples above for continuous compound interest. This video
illustrates how to calculate the amount needed to invest when given
how much one earns.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 15 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Read Section 10.5 in Chapter 10 of your textbook,
pages 410–413, to learn about logarithmic functions. The logarithmic
functions are the inverse functions to the exponential. Thus, these
functions become important when needing to solve an exponential
equation with the unknown in the exponent. Logarithmic functions
have many real-world applications. When the magnitude of an
earthquake is reported, the report does not give the actual tremor,
since this is a very large number. The report gives the log of the
tremor. The same is true for reporting sound (decibels) and acidity
(ph). Note that this reading also covers the topics in Subunits
5.7.1–5.7.3.

Reading this section and taking notes should take approximately 2
hours.

Instructions: Complete pages 112–114 of Wallace’s workbook. Try to
complete this exercise before watching the videos in Subunits
5.7.1–5.7.3, and then review the worksheet as you follow along with
the videos for solutions.

Instructions: Watch the video linked above, which discusses how to
convert from a logarithmic expression (log) to an exponential
expression and vice versa. The conversion formula is: bx
= a is equivalent to logba = x.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.7.2 Evaluating LogarithmsNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.7.

Instructions: Watch the video linked above, which discusses
evaluating logarithmic expressions. For general bases (bases other
than 10 and e which are on all scientific calculators), determining
the value of a log may be easier to convert to an exponential
expression. Then you can use previous learned techniques to solve
the exponential equation. Thus, convert logba = x to
bx = a. Then use the techniques you learned in Subunit
5.5.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

5.7.3 Solving Logarithmic EquationsNote: This subunit is also covered by the reading and assessment
assigned in Subunit 5.7.

Instructions: Watch the video linked above, which discusses solving
logarithmic equations. As in the previous video, solving logarithmic
equations is easier if first converted to an exponential equation.

You may watch the video as often as you please. You may refer to
the video when doing the assessment if necessary.

Watching this video and pausing to take notes should take
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Final Exam
- Final Exam: The Saylor Foundation‘s “MA004 Final Exam”
Link: The Saylor Foundation‘s “MA004 Final
Exam”

Instructions: You must be logged into your Saylor Foundation School
account in order to access this exam. If you do not yet have an
account, you will be able to create one, free of charge, after
clicking the link.